BC 


UC-NRLF. 


PRIMER  o/  LOGIC 


SMITH 


GIFT   OF 


Primer  of  Logic 


BY 


HENRY  BRADFORD  SMITH 

Assistant  Professor  of  Philosophy  in  the 
University  of  Pennsylvania 


B.  D.  SMITH  A  BROS. 

PULASKI,  VA. 

1917 


PREFACE 

In  the  pages  which  follow  will  be  found  the  outlines 
of  the  logic  of  a  set  of  categorical  forms  which  are  not 
Aristotle's  own.  It  is  only  the  fragment  of  a  general 
theory,  but  the  content  of  all  the  chapters  of  the  old  logic, 
which  are  commonly  regarded  as  essential,  except  that 
one  which  deals  with  the  calculus  of  classes,  will  be  found 
included.  In  the  first  appendix  the  relation  of  these  new 
forms  to  the  traditional  ones  has  been  pointed  out  in  detail. 

I  have  at  least  one  debt,  which  calls  for  a  definite 
acknowledgement.  It  is  to  my  friend  and  teacher,  Mr. 
Edgar  A.  Singer,  Jr.,  that  I  owe  what  training  I  have  had 
in  the  science.  He  has  never  failed,  through  hints  thrown 
out  in  conversation,  to  correct  my  misapprehensions. 
But  my  indebtedness  is  more  specific  than  this.  I  have  so 
far  employed  his  own  method,  a  method  developed  in  his 
academic  lectures,  that  I  could  scarcely  have  ventured 
upon  the  publication  of  these  outlines  of  a  theory  without 
his  express  permission. 

H.  B.  S. 


TABLE  OF  CONTENTS 

CHAPTER    I  PAGES 

§1-3.      The  fundamental  properties  of  the  categorical 

forms.    Exercises 1-9 

CHAPTER    II 

§4-8.  The  relationships  of  "better"  and  "worse." 
Deduction  of  the  moods  of  immediate  infer- 
ence and  of  the  syllogism  by  rule.  Exercises. .  .  10-17 

CHAPTER  III 

§9-14.  The  relationships  of  "better"  and  "worse" 
defined.  Symbolic  deduction  of  the  moods 
of  immediate  inference  and  of  the  syllogism. 
Exercises 18-27 

CHAPTER   IV 
§15.        The  general  solution  of  the  sorites.     Exercises..  .28-34 

APPENDIX   I 

On  the  simplification  of  categorical  expression 

and  the  reduction  of  the  syllogistic  figures 35-44 

APPENDIX    II 

Historical  note  on  De  Morgan's  new  preposi- 
tional forms  . .  . .  45-48 


CHAPTER    I 

§1.  In  1846  Sir  William  Hamilton  published  the 
prospectus  of  an  essay  on  a  "New  Analytic  of  Logical 
Forms,"*  which  revived  the  question  as  to  whether  or 
not  the  quantity  of  the  predicate  of  the  categorical  forms 
shquld  be  stated  explicitly.  The  chief  difficulties  of  his 
system  result  from  the  ambiguity  of  the  meaning  of  some, 
from  the  impossibility  of  making  every  form  of  categorical 
expression  simply  convertible,  and  from  the  seemingly 
curious  effort  to  establish  an  order  of  better  and  worse 
between  the  relations  connecting  subject  and  predicate. 
Four  of  Hamilton's  eight  forms  are  redundant.  The  four 
that  remain  will  be  represented  here  by  the  letters,  a, 

ft  7,  e- 

Accordingly  let 

aab  =  all  a  is  all  b, 
/3ab  =  some  a  is  some  b, 
7ab  =  all  a  is  some  b, 
eab=no  a  is  b. 

Here  some,  the  some  expressed  explicitly  in  ft  and  7, 
means  some  at  least,  not  all.  This  meaning  of  the  word 
is  established  unambiguously  by  the  properties  of  the 
forms.  In  addition  to  these  abbreviations  we  will  employ 
the  notation: 

Xab  =xab  (is  true), 
x'ab=xab  (is  false), 

xab  '  y&b    =Xab  (is  true)  and  yab  (is  true), 
Xab  +  y&b    =xab  (is  true)  or  yab  (is  true), 
xab  Z  yab    =xab  (is  true)  implies  yab  (is  true), 
(xab  Z.  yab)'  =xab  (is  true)  does  not  imply  yab  (is  true). 

*Lectures  on  Logic,  ed.  by  Mansel  and  Veitch,  Boston, 
Gould  and  Lincoln,  1863. 


2  A  PRIMER  OF  LOGIC 

§2.  The  implications,  which  are  given  below,  express 
the  chief  characteristics  of  the  forms.  The  theorems 
follow  by  the  principle  of  the  denial  of  the  consequent, 
which  may  be  written  in  the  two  forms: 

(x  Z  y')  Z  (y  Z  x')  and  (x'  Z  y)'  Z  (y'  Z  x)'. 

Postulates:* 

ttab  Z   0'ab  0ab  Z   7'ab  (o'ab  ^    £ab)'  (0'ab  /  Tab)' 

dab  Z  T'ab  /?ab  Z     e'ab  (d'ab  Z  Tab)'  (jS'ab  ^     Cab)' 

«ab  Z     e'ab  7ab  Z     e'ab  (tt'ab  Z     €ab)'  (7'ab  ^     Cab)' 

7ab  Z  7'ba  (7'ab  Z  7ba)' 

Theorems: 

€abZa'ab  7abZd'ab  (c'abZttab)'  (7'ab  Z    aab)' 

€ab  Z    jS'ab  7ab  Z   jS'ab  (  «'ab  /    jSabX  (7'ab  Z    jSab)' 

Cab  Z  7'ab  0ab  Z    a'ab  (e'ab  ^   Tab)'  (0'ab  Z    ttab)' 

Let  us  postulate  in  addition  that: 

aab  Z  (a'ab)'  (o'ab)'  ^  aab 

/3ab  Z   (/3'ab)'  O'l*)'  ^   £ab 

7ab  Z  (T'ab)'  (7'.b)'  ^  Tab 

6abZ  (e'ab)'  (c'ab)'Z   eab 

Then,  if  k^rjiWafo  and  if  kab  and  wab  represent  only 
the  unprimed  letters,  aab,  /3ab,  7ab,  eab,  a  complete  induction 
of  the  propositions  given  above  yields  the  general  result: 

kabZ    (k'ab)',       (k'ab)'Zkab,  I 

kab  Z   W'ab,  (W'ab      Z   kab)',  II 

*These  assumptions  are  in  accord  with  those  of  the  common 
logic,  but  no  longer  hold  when  the  terms  are  allowed  to  take  on 
the  limiting  values  0  and  1 ;  for  7OI  and  e0i  are  both  true  prop- 
ositions. The  assumptions  (7abZ  e'ab)'  and  (eabZ  7'ab)  will 
be  characteristic  of  a  more  general  logic,  which  will  include 
the  classical  logic  as  a  special  case.  (See  the  concluding  remarks 
of  chapter  III.) 


A  PRIMER  OF  LOGIC 

and  by  denial  of  the  consequent, 

(xZ  y)  Z(y'Z  x'), 
(xZy)'Z(y'Zx')', 

it  follows  that: 

(k'ab)'Z   W'ab,       (w'abZ    (k'ab)')'.  HI 

If  now  we  postulate: 

(dab  Z    tt'ab)'  (a'ab  /    aab)' 

(/3ab  Z   /3'ab)'  (/3'ab^0ab)' 

(7ab^7'ab)'  (7'ab/7ab)' 

(eab^    €'ab)'  (€'ab/    eab)' 

and,  consequently, 

(kab  /   k'ab)',       (k'ab/kab)', 

it  will  follow  by* 

(xZ  y)  (xZ  z)'Z(yZ  z)', 

(xZz)/(yZz)Z(xZy)/, 
that  (kab  Z  (w'ab)')',    ((w'ab)'  Z  kaby.  IV 


Finally  if  we  assume 

dab  £    dba>     ^ab  -^   ^ba»     Tab  Z  7&b,      Cab 

it  will  follow,  by 

(x  Z  y)  (y  Z  z)  Z  (x  Z  z)  and  (x  Z  y)  Z  (y'  Z  x'), 
that 

kab  /   kab,       k'ab  Z   k'ab. 

Definitions. 


If  xab  Z  y'ab  and  y'ab  Z  xab,  xab  is  said  to  be  contra- 
dictory to  yab-  By  I,  kab  is  contradictory  to  k'ab  and,  by 
I',  k'ab  is  contradictory  to  kab. 

"These   are   obtained    from   (x  Z  y)  (y  Z  z)  Z  (x  Z  z)  by 
(xy  Z  z)  Z  (xz;  Z  yO  and  xy  Z  yx. 


4  A  PRIMER  OF  LOGIC 

If  xab  Z  y'ab  and  (y'ab  Z  xab)',  xab  is  said  to  be  contrary 
to  yab.  By  II,  kab  is  contrary  to  wab,  and  conversely, 
since  kab  and  wab  are  interchangeable. 

If  (xab  Z  y'ab)'  and  y'ab  Z  xab,  xab  is  said  to  be  subcon- 
trary  to  yab.  By  III,  k'ab  and  w'ab  are  subcontrary  pairs. 

If  (xab  Z  y'ab)'  and  (y'ab  Z  xab)',  xab  is  said  to  be  sub- 
alternate  to  yab.  By  IV,  k'ab  and  wab  are  subalternate 
pairs. 

§3.  Having  classified  the  categorical  forms  under 
these  heads,  it  remains  to  differentiate  them  by  means  of 
their  formal  properties.  If  we  assume  as  valid, 

a  aa  Z  aaa,      e  aa  Z   eafi,     (aaa  Z  a  aa)  ,      (€aa  Z   e  aa)  , 

where  a  represents  the  class  contradictory  to  a,  (non-a), 
the  other  propositions  given  below  may  be  derived.* 

a  aa  Z    aaa  (ctaa     Z    0,  aa)  tt  aa  Z    tt  aa  (tt  aa  Z    O,aa) 

0aa    Z   /3'aa  (/3'aa  Z    £„)'  ft  afi  Z   iS'a,  (^afi  Z    /3aa)'          V 

7aa    Z  T'aa  (Vaa  Z  Taa)'  7  aa  Z  ^aa  (Vafi  Z  7aa)' 

€aa    Z     e'aa  (e'aaZ     6aa)'  e'aa  Z     €aa  ( €aa  Z     e'aa)' 

The  results  of  V,  together  with  the  non-convertible 
character  of  7**,  are  enough  to  establish  the  definitions 
of  the  four  forms. 


*Under  the  conditions  mentioned  above,  in  note,  p.  2,  we 
shall  have  to  write  (7aa  Z  7'aa)'.  Implications  V  are  an  extension 
of  the  meaning  of  implication,  made  necessary  by  our  having 
to  call  aaa  and  eaa  true  propositions.  (See  Boole,  Investigation 
of  the  Laws  of  Thought,  chap.  XI,  p.  169.) 

**The  operation  of  simple  conversion  consists  in  interchang- 
ing subject  and  predicate.  By  the  principle,  (xZ  z/(yZ  z)  Z 
(xZ  y)',  and  what  has  gone  before,  we  have: 

(TabZ  Vab/CTbaZ  T'ab)  Z  (Tab  Z  Tba)'. 


A  PRIMER  OF  LOGIC  5 

Definitions. 

A  form  which  is  the  contrary  of  itself  is  called  a  null- 
form.  By  V,  0aa,  Taa,  caa,  aa&,  /3aa,  7&a,  are  null-forms. 

A  form  which  is  the  subcontrary  of  itself  is  called  a 
one-form.  By  V,  aaa,  eaa,  are  one-forms. 

If  xab  is  unprimed  and  xaa  a  one-form  then  xab  is 
called  an  a-form. 

If  xab  is  unprimed  and  simply  convertible  and  if 
xaa  and  xaa  are  null-forms,  then  xab  is  called  a  /3-form. 

If  xab  is  unprimed  and  not  simply  convertible,  then 
xab  is  called  a  7-form. 

If  xab  is  unprimed  and  xaa  a  one-form,  then  xab  is 
called  an  €-form. 


A  PRIMER  OF  LOGIC 


EXERCISES 

(1)     Assuming  kab  =  kab  *  kab,  kabZ  w'ab,  show  that, 

OfabZ   j'ab  7'ab       ab   T'ba 


aab  7ab  eab  7ba 
7abZ  a'ab  /3'ab  e'ab  7'ba 
eab  Z  a'ab  |8  ab  7  ab  7  ba 
by  the  aid  of, 

(xZy)  (yZz)Z(xZz), 
(xZy)Z(zxZzy). 

Equality  is  defined  by 

(xZy)(vZx)Z(x  =  y), 
(x  =  y)Z(xZ  y)  (yZ  x). 

If  now 

0'ab   7ab    c'ab   T'ba  Z  «ab 


^ 

a^ab   ^'ab    C^ab  Vba  Z  7ab 
«  ab    jS'ab  7  ab   7'ba  Z    €ab 

it  will  follow  that 

aab  =  /^ab  T^b    €^ab  7^ba  «^ab=/3ab  +  7ab  +    eab  +  7ba 

j8ab  =«'ab  T'ab    «'ab  7'ba  /3'ab  =«ab  +  7ab  +    €ab  +  7ba 

7ab=«'ab  ^ab    €^ab  7^ba  7/'ab=«ab  +  /3ab  +    €ab  +  7ba 

6ab  =  «'ab  j8  ab  7  ab  T  ba  e'ab  =QJab  +  /3ab  +  7ab  +  7ba 

The  second  set  of  equations  follows  from  the  first  by  the 
principle,  that  the  contradictory  of  a  product  is  the  sum  of  the 
contradictories  of  the  separate  factors,  and  by  substituting  kab 
directly  for  (k'ab)'. 

(2)  Show  by  the  method  of  the  last  example  that 

aab=«ab  /3'ab=aab  7'ab=«ab    c'ab 

0ab  =  /3ab  a'ab  =  /3ab  7'ab  =  |8ab  e'ab 

7ab=7ab  ^ab=7ab  ^ab  =  7ab    e'ab 
Cab  =  Cab  «'ab  =  «ab  ^'ab  =  €ab  7'ab 

Show  too  that 

«ab=«ab  0'ab  7'ab=«ab  j^'ab    t/ab=«ab  ?'ab     c'ab,  etc.,    etc. 

and  that  aab=aab  /3'ab  7'ab  c'&b,  etc.,  etc. 
Derive  the  analogues  of  the  first  set: 

a'ab  =  C/ab  +  /3ttb  =  «'ab  +  Tab  =«'ab  +   Cabi  etc.,  etc, 


A  PRIMER  OF  LOGIC  7 

Accordingly,  since  kab  =  kab  '  w'ab  and  k/ab  =  k/ab  +  wab,  any 
primed  letter  is  a  modulus  of  multiplication  with  respect  to 
any  unprimed  letter  not  itself  and  any  unprimed  letter  is  a 
modulus  of  addition  with  respect  to  any  primed  letter  not  itself. 

The  prepositional  zero  is  defined  by 

(xyZ  (xy)')Z(xyZ  0), 
(xyZ  0)Z(xyZ  (xy)'), 

and  the  prepositional  one  by 

((xy/Zxy^lZxy), 
(1Z  xy)Z((xy)'Z  xy). 

From  the  principles, 

(xZ  y')Z(xyZ  (xy)'), 
(xyZ(xy)')Z(xZy'), 
it  follows  that 

(xZyOZ(xyZO) 
(xjZO)Z(xZ  y') 

(3)  Derive: 

kab  '  k'ab  /  (kab  '  k'ab)';  (kab  +  k'ab)'  Z  kab  +  k'ab; 
kab  '  wab  Z  (kab  '  wab)';  ((kab  +  wab)'  Z  kab  +  wab)'; 
(k'ab  '  w'ab  Z  (k^  '  w'ab)')';  (k7ab  +  w'ab)'  Z.  k'ab  +  w'ab; 
(kab  *  w'ab  ^  (kab  '  w'ab)') ';  ((kftb  +  wf&bYZ.  kab  +  w'.b)'. 

(4)  Show  that 

«ab  |8ab  ^   0,  «ab  7ab  Z   0,  etc. 

1  Z  a'ab  +  0'ab,  1  ^  a'ab  +  7'ab,  etc. 
and  that 

«'ab  #'ab  T'ab    «'ab  T'ba  -^   0 

1  Z  a:'ab  +  j3'ab  +  7'ab  +  e'ab  +  T'ba 

(5)  Assuming  xab  =  xab  '  xab,  xab  =  xab  +  xabf 
show  that 

a'ab  ^'ab=7ab  +   «ab  +^7ba,  «'ab  7'ab  =  /3ab  +    «ab  +  7ba,  CtC. 
«ab  +  /3ab=7'ab    C  ab  7'ba,  «ab  +  7ab  =  /3'ab    c'ab  7'ba»  etc. 

(6)  Derive  the  general  result: 

(ka.b/   Wa,b)'. 

The  comma  between  the  terms  indicates  that  the  term  order  is 
not  fixed.    Thus  ka,b  stands  for  either  kab  or  kba. 


8  A  PRIMER  OF  LOGIC 

(7)  From  the  principle, 

(xZz)'(yZz)Z(xZy)', 
and  the  postulate  (aaa  Z  a'aa)', 
derive  (aaaZ  /3aa)',  («aaZ  Taa)',  («aaZ  eaa)'. 

(8)  From  the  principles, 

(xZy)(yZz)Z(xZz), 
(x'Z  x)Z(yZ  x), 
(xZxOZ(xZ  y), 

and  the  postulate,  c/aaZ  aaa,  show  that  all  propositions  of  the 
form  xaaZ  yaa,  except  the  three  cases  in  the  last  example,  are 
valid  implications,  xaa  and  yaa  representing  only  the  unprimed 
letters. 

(9)  By   the   method    of    the    last    example,    prove    that 
(aaa  Z  a'aay  is  the  only  invalid  implication  of  the  form  xaa  Z  y'aa. 

(10)  Derive  seven  valid  implications    in  each  one  of    the 
forms  x'aa  Z  yaa  and  x'aa  Z  yaa  and  nine  invalid  implications  of 
each  one  of  the  same  forms. 

(11)  From  (€aa  Z  e'aa)'  by  the  method  of  example  7  derive 
(eafl  Z  aaa)',  (€aa  Z  0aa)',  (caa  Z 


(12)  From   e'aa  Z  €a5  by  the  method  of  example  8  derive 
thirteen   valid   implications. 

(13)  Show  that  (6aa  Z  e'aa)'  is  the  only  invalid  implication 
of  the  form  xaa  Z  y'aa. 

(14)  Derive  the  following  implications: 


iz«aa 

0/aaZ   0 

1  Z  a'aa 

«aa  Z  0 

1  Z   0'aa 

j8aa  Z  0 

1  Z  j8'aa 

/3aa  Z  0 

izyaa 

Taa    Z   0 

1  Z  T'aa 

Taa  Z  0 

1Z   e'aa 

6aa    Z   0 

1Z   eaa 

€'afiZ    0 

(lZo/aay 

(«aa    /   0)' 

(izaaay 

(«'aaZO)' 

03'aazoy 

az/3aay 

tf'aaZ  0)' 

(IZT.a)' 

(VaaZO)' 

(14  7s.) 

(T'aaZO)' 

(1Z     6aa)' 

(e'aaZO)' 

(1Z     6^)' 

(eaaZ  0)' 

Some  of   the   postulates  in    the  text   (p.  2)  of   the   form, 
(k'abZ  wab)',  (kabZ  k'ab)',  (k'ab  Z  kab)',  may  be  established  by 


A  PRIMER  OF  LOGIC  9 

reducing  them  to  one  of  the  forms,  (k'aaZ  waa)',  (k'aaZ  waa)' 
(kaaZ  k'aa)',  etc.,  cases  already  considered  in  preceding  exercises 

(15)  Establish  the  invalidity  of 

«'ab  Z   £ab  «'ab  Z  Tab  Vab  ^  Tba 

/3'ab^  Tab  /3'ab^     €ab  Vab  ^    «ab 

«'ab  ^  «ab  jS'ab  ^    l^ab  V ab  ^   Tab 

'ab  €ab  Z    c'ab  C  ab  Z    €ab 


CHAPTER    II 

§4.  At  this  point  in  our  theory  it  will  be  necessary 
to  introduce  certain  indefinables,  which  we  shall  call  the 
distinctions  of  better  and  worse,  following  a  suggestion  of 
Sir  William  Hamilton's.*  For  our  immediate  purpose 
it  will  be  enough  to  define  better  than  and  worse  than  de- 
notatively, establishing  an  order  among  the  four  forms  by 
a  simple  enumeration.  Better  than  and  worse  than  are  not 
transitive  relations.  When  we  wish  to  express  the  rules 
for  the  deduction  of  the  moods  symbolically,  we  shall 
have  to  invent  symbols  to  represent  worse  than  (/),  doubly 
worse  than  (//),  and  trebly  worse  than  (///).  This  necessity 
is  avoided  in  the  verbal  statement  of  the  principles  of  de- 
duction by  the  words  "in  the  same  degree"  (see  rule  1 
below). 

Definitions. — An  6-form  is  worse  than  an  a-,  a  0-,  or 
a  7-form. 

A  7-form  is  worse  than  an  a-  or  a  /3-form. 
A  /3-form  is  worse  than  an  a-form. 

Best.     a~  /3--  7—  e     Worst. 

§5.  Immediate  inference  is  a  form  of  implication 
belonging  to  one  of  the  types: 

1.  xab/  yab-     2.  xabZ  yba. 

These  differences  are  known  as  the  first  and  the  sec- 
ond figures  of  immediate  inference  respectively. 

The  part  to  the  left  of  the  implication  sign  is  called 
the  antecedent;  the  part  to  the  right  is  called  the  consequent. 

*Lectures  on  Logic,  Appendix,  p.  536. 


A  PRIMER  OF  LOGIC  11 

Since  x  and  y  may  take  on  any  of  the  forms,  a,  /3, 
7,  c,  there  will  be  sixteen  propositions  of  each  type,  ob- 
tained from  the  permutations  of  the  letters  two  at  a  time 
and  by  taking  each  letter  once  with  itself.  Each  one  of 
the  sixteen  distinct  propositions  in  each  one  of  the  two 
figures  is  called  a  mood  of  immediate  inference.  The 
rules  which  follow  below,  applied  to  the  postulates,  will 
yield  all  the  true  and  all  the  false  propositions  of  each  type. 

Valid  Moods. 

1.  In  any  valid  mood  of  the  first  figure  make  ante- 
cedent and  consequent  worse  in  the  same  degree. 

2.  In  any  valid  mood  convert  simply  in  any  form 
but  7. 

Postulate:     aab  Z.  aab.  Theorems:    The  other  (6)  valid 

moods. 
Invalid  Moods. 

1.  In  any  invalid  mood  of  the  first  figure  make 
antecedent  and  consequent  worse  in  the  same  degree. 

2.  In  any  invalid  mood  of  the  first  figure  interchange 
antecedent  and  consequent. 

3.  In  any  invalid  mood  convert  simply  in  any  form 
but  7. 

Postulates:* 

{aab/0ab}';     {dab/Tab}';     (aab/     Cab}';     {7ab/7ba}'. 

Theorems:    The  other  (21)  invalid  moods. 

§6.    We  may  also  formulate  rules  for  the  detection 
of  the  invalid  moods.     These  are: 


*The  mark  0  over  the  bracket  is  interred  to  indicate  that 
the  mood  is  invalid, 


12  A  PRIMER  OF  LOGIC 

1.  If  the  antecedent  be  worse  than  the  consequent, 
the  mood  is  invalid. 

2.  If  the  antecedent  be  better  than  the  consequent, 
the  mood  is  invalid. 

Definition. — Distributed  terms  are  those  modified, 
either  implicitly  or  explicitly,  by  the  adjective  all,  i.  e. 
the  subject  of  the  a-,  7-  and  e-form,  and  the  predicate  of 
the  a-  and  e-form. 

3.  it  a  term  be  distributed  in  the  consequent  but 
undistributed  in  the  antecedent,  the  mood  is  invalid. 

§7.  A  syllogism  is  a  form  of  implication  belonging 
to  one  of  the  types: 

1.  xbaycbZ  zca=(xy  z)x 

2.  xabycbZ  zca=(xy  z)2 

3.  xba  ybc  Z  zca  =(x  y  z)3 

4.  xabybcZ  zca=(xyz)4 

These  differences  are  known  as  the  first,  second,  third, 
and  fourth  figures  of  the  syllogism  respectively.  The  two 
forms  conjoined  in  the  antecedent  are  called  the  premises 
and  the  consequent  is  called  the  conclusion.  The  predi- 
cate of  the  conclusion  is  called  the  major  term  and  points 
out  the  major  premise,  which  by  convention  is  written 
first,  and  the  subject  of  the  conclusion  is  called  the  minor 
term  and  points  out  the  minor  premise.  The  term  common 
to  both  premises  and  which  does  not  appear  in  the  con- 
clusion is  called  the  middle  term. 

Since  x,  y  and  z  may  have  any  one  of  the  values, 
a,  ft  7,  e,  there  will  be  sixty-four  ways  in  each  one  of  the 
four  figures,  called  the  moods  of  the  syllogism,  in  which 
x  y  Z  z  can  be  expressed.  There  will  be  consequently 


A  PRIMER  OF  LOGIC  13 

two  hundred  and  fifty-six  cases  to  consider.  Twenty- 
nine  of  these  are  valid  implications;  the  remaining  two 
hundred  and  twenty-seven  are  invalid.  From  the  rules 
and  postulates  below,  all  the  moods,  valid  and  invalid, 
may  be  deduced. 

Valid  Moods. 

1.  In  any  valid  mood  of  the  third  figure  make  a  like 
major  premise  and  conclusion  worse  in  the  same  degree. 

2.  In  any  valid  mood  of  the  second  figure  make  a 
like  minor  premise  and  conclusion  worse  in  the  same  degree. 

3.  In  any  valid  mood  convert  simply  in  any  form 
but  7. 

Postulates  :  Theorems  : 

(aaa)  x  (aaa)2,3(4  (/3aj8)If2f3i4 


(7^)2,4 

Invalid  Moods. 

1.  In  any  invalid  mood  of  the  third  figure  make  a 
like  major  premise  and  conclusion  better  in  the  same  degree. 

2.  In  any  invalid  mood  of  the  second  figure  make 
a  like  minor  premise  and  conclusion  better  in  the  same 
degree. 

3.  If  the  premises  and  conclusion  of  an  invalid  mood 
in  the  fourth  figure  are  all  alike,  make  them  all  worse  in 
the  same  degree. 

4.  If  the  premises  and  conclusion  of  an  invalid  mood 
are  all  alike  make  the  conclusion  any  degree  better  or  any 
degree  worse. 


14  A  PRIMER  OF  LOGIC 

5.  If  the  premises  and  conclusion  of  an  invalid  mood 
are  all  unlike,  interchange  them  in  any  order. 

6.  In  any  invalid  mood  convert  simply  in  any  form 
but  7. 

Postulates:* 

(aa/3)',         (tfc)',          tfry)',          (777)'.         W, 

(aa«)', 
(aPy)', 


Theorems: 
The  other  (208)  invalid  moods. 

§8.  As  in  the  case  of  immediate  inference  we  may 
formulate  rules  for  the  detection  of  the  invalid  moods  of 
the  syllogism.  These  are  five  in  number. 

1.  A  mood  is  invalid  if  the  conclusion  differ  from  the 
worse  premise. 

2.  A  mood  is  invalid  if  an  a-  and  a  7-  premise  be 
conjoined  in  the  antecedent  and  the  middle  term  be  un- 
distributed in  the  major  premise. 

3.  A  mood  is  invalid  if  the  middle  term  be  undis- 
tributed   in    both    premises. 

4.  A  mood  is  invalid  if  a  term  which  is  distributed 
in  the  conclusion  be  undistributed  in  the  premise. 

5.  A  mood  is    invalid  if  each   premise  be    in    the 
e-form.** 

*The  mark  (')  over  the  bracket  is  intended  to  indicate  that 
the  mood  is  invalid. 

**These  rules  are,  of  course,  not  sufficient  to  declare  (766)2,4 
and  (€7c)I)2  invalid,  in  case  we  decide  to  so  regard  them.  See 
the  concluding  remarks  of  chap.  III. 


A  PRIMER  OF  LOGIC  15 


EXERCISES 

If  xa,b^  ya,b  be  denoted  simply  by  (xy),  (the  comma  be- 
tween the  terms  indicating  that  the  term  order  and  so  the  figure, 
is  not  determined),  the  array  of  sixteen  propositions  may  be 
constructed  thus: 

aa  fia  yy.  ea 

"^  00  70  $ 

or?  07  77  €7 

ae  j8c  76  ee 

the  moods  valid  in  both  figures  being  underlined  twice,  the  one 
valid  only  in  the  first  figure  being  underlined  once.  Applying 
the  first  rule  to  the  postulate,  we  obtain  in  succession,  j3/3,  77, 
ee,  in  the  first  figure;  and  converting  simply  in  the  consequent 
of  those  valid  in  the  first  figure,  except  77,  we  obtain  aa,  /3/3,  ce 
in  the  second  figure. 

(1)  From  the  rules  and  postulates  for  the  derivation  of  the 
invalid  moods  deduce  the  remaining  twenty-one  invalid  moods. 

The  rules  for  the  immediate  detection  of  the  invalid  moods 
are  all  necessary  if  we  can  point  to  at  least  one  example  which 
falls  uniquely  under  each  rule.  They  are  sufficient  if  they  de- 
clare all  the  invalid  moods  to  be  invalid. 

(2)  Construct  the  set  of  propositions  of  immediate  inference 
and  place  after  each  invalid  mood  the  number  of  a  rule  which 
declares  it  to  be  invalid. 

(3)  Make  a  list  of  moods  which  are  declared  invalid  by  the 
first  rule  and  by  no  other  rule,  and  a  list  of  moods  which  are 
declared  invalid  by  the  second  rule  and  by  no  other  rule. 

|) 

(4)  Find  an  invalid  mood  which  is  declared  invalid  by  the 
third  rule  and  by  no  other  rule  and  prove  that  it  is  the  only 
unique  illustration  of  this  rule. 

For  those  who  approach  the  study  of  the  syllogism  for  the 
first  time,  it  may  facilitate  manipulation  to  point  out  the  general 
effect  of  conversion  in  the  form  of  certain  rules. 

1,  Simple  conversion  in  the  major  premise  changes  the 
first  figure  to  the  second  and  conversely,  the  third  figure  to  the 
fourth  and  conversely. 


16  A  PRIMER  OF  LOGIC 

2.  Simple  conversion  in  the  minor  premies,  changes  the 
first  figure  to  the  third  and  conversely,  the  second  figure  to  the 
fourth  and  conversely. 

3.  Simple  conversion  in  the  conclusion  changes  the  first 
figure  to  the  fourth  and  conversely  and  leaves  the  second  and 
third  figure  unchanged. 

It  must  of  course  not  escape  the  beginner's  notice  that  the 
effect  of  converting  simply  in  the  conclusion  is  to  interchange  the 
premises,  since  the  major  term  then  becomes  the  minor  term  and 
the  minor  term  becomes  the  major  term.  The  conjunctive 
relation  of  logic  being  commutative,  the  order  of  the  premises 
is  indifferent,  but  we  agree,  as  a  matter  of  convention,  always 
to  write  the  major  premise  first. 

(5)  From  (eye)i,  and  the  third  rule  under  the  valid  moods 
alone,  deduce  (eye)2,  (7ee)2  and  (yee)4. 

(6)  From  the  rules  and  postulates  deduce  the  remaining 
valid  moods. 

(7)  Assuming  only  the  third  rule  under  the  valid  moods 
and  the  rule  :  in  any  valid  mood  of  the  first  figure  make  a  like 
major  premise  and  conclusion  worse  in  the  same  degree,  deduce 
all  the  remaining  valid  moods  from  (aaa)z,  (777)1  and  (077)1. 

(8)  Assuming  only  the  second  and  third  rules  under  the 
valid  moods  deduce  the  remaining  valid  moods  from  (aaa)lt 
(777)i,  (7<*7)i  and 


(9)  From  (efieYt  alone  deduce  seventy-eight  other  invalid 
moods. 

(10)  From  (afieYt  alone  deduce  twenty-three  other  invalid 
moods. 

(11)  Deduce  the  invalid  moods  in  the  first  figure  which  have 
a  7-minor  premise. 

The  rules  for  the  immediate  detection  of  the  invalid  moods 
are  sufficient,  if  they  declare  all  the  moods  not  already  found 
to  be  valid  to  be  invalid.  They  are  all  necessary  if  we  can  point 
to  at  least  one  example  which  falls  uniquely  under  each  rule. 

(12)  Construct  the  array  of  the  syllogism  and  place  after 
each  invalid  mood  the  number  of  a  rule  that  declares  it  to  be 
invalid, 


A  PRIMER  OF  LOGIC  17 

(13)  Show  that  it  follows  from  one  of  the  rules  alone  that 
two  jS-premises  do  not  imply  a  conclusion. 

(14)  Prove  that  there  are  only  two  moods  which  illustrate 
the  second  rule  uniquely. 

(15)  Make  a  list  of  examples  which  fall  uniquely  under 
each  one  of  the  rules. 


18  A  PRIMER  OF  LOGIC 


CHAPTER    III 

§9.  In  this  third  chapter  it  is  proposed  to  completely 
define  the  relationships  of  better  and  worse  by  deducing  all 
the  true  and  all  the  false  propositions  into  which  these 
relationships  may  enter  and  then  to  give  a  complete  ex- 
pression in  the  language  of  symbols  of  the  rules  for  the 
deduction  of  the  moods  of  immediate  inference  and  the 
syllogism. 

§10.  Let  us,  first  of  all,  invent  symbols  to  denote 
worse  than,  doubly  worse  than,  and  trebly  worse  than,  i.  e. 

x   /  y  =x  is  worse  than  y, 

x  //  y  =x  is  doubly  worse  than  y, 

X///Y  =x  is  trebly  worse  than  y, 

and  let  us  add  the  following: 

Definition. — In  the  propositions,  x  /  y,  x//y,  and 
x///y,  x  is  called  the  inferior,  y  the  superior  form. 

Since  x  and  y  may  take  on  any  of  the  four  forms 
a,  /3,  7,  e,  there  will  be  sixteen  possible  propositions  of  each 
type,  x  /  y,  x  //y  and  x///y,  obtained  by  permuting  the 
letters  two  at  a  time  and  by  taking  each  letter  once  with 
itself.  The  following  postulates  and  principles  will  yield 
all  the  valid  moods  of  each  type.  We  have  assumed  four 
principles  here  because  the  principles  for  the  deduction  of 
the  invalid  moods  may  be  derived  from  these  four  as 
theorems. 

Principles: 

i.  (x  /  y)  (y  //z)  Z  (x///z)     iii.  (x  /  z)  (y///z)  Z  (y  //x) 
ii.  (x  /  y)  (z  //x)  Z  (z///y)     iv.  (x  /  z)  (y  //  z)  Z  (y  /  x) 

Postulates:    /3/a;     6/7;    7//a. 

Theorems:     7/0;     e//0;     e///a. 


A£PRIMER  OF  LOGIC  w 

We  may  also  formulate  rules  for  the  derivation  of 
the  moods.  It  will  then  be  necessary  to  assume  one  pos- 
tulate only. 

Definition: — In  the  propositions,  x  /  y,  x  //y  and 
x///y,  the  relation  connecting  x  and  y  is  known  as  the 
worse-relation. 

Definition: — Trebly  worse  (///)  is  worse  than  doubly 
worse  (//)  and  doubly  worse  than  worse  (/).  Doubly 
worse  (//)  is  worse  than  worse  (/). 

The  rules  are: 

1.  In  any  valid  mood  make  superior  and  inferior 
form  one  degree  better  or  one  degree  wrorse. 

2.  In  any  valid  mood  make  inferior  form  and  worse- 
relation  one  degree  better  or  one  degree  worse. 

Postulate :  Theorems  : 

/3   /  a.  The  other  (5)  valid  moods. 

The  invalid  moods  of  each  type  may  be  derived  from 
the  following  postulates  and  principles: 

Principles:* 

i.     (x   /  y)    (x///z)'Z(y//z)' 

(x///z)'(y//z)  Z(x  /  y)' 
ii.  (x  /  y)  (z///y)'Z(z//x)' 

(z///y)'  (z//x)  Z(x  /  y)' 
iii.  (x  /  z)  (y//x)'Z(y///z)' 

(y//x)'(y///z)  Z(x   /  z)' 

iv.     (x  /  z)    (y   /  x)'Z(y//z)' 
(y  /  x)'(y//z)  Z(x    /  z)' 

These  principles  follow  from  those  used  for  the  deduction 
of  the  valid  moods  by  (xyZ  z)  Z  (xz'Z  y'). 


20  A  PRIMER  OF  LOGIC 

Postulates:  Theorems: 

(a/VAX;  (0//A)';  (7///e)';  The  other  (3  6)  invalid  moods. 

(€///€)';  (e///7)'; 


As  in  the  case  of  the  valid  moods,  rules  may  be  form- 
ulated for  the  derivation  of  the  invalid  moods.  Here  it 
will  be  necessary  to  assume  only  three  postulates.  The 
rules  are: 

1.  In  any  invalid  mood  make  superior  and  inferior 
form  one  degree  better  or  one  degree  worse. 

2.  In  any  invalid  mood  make  inferior  form  and  worse- 
relation  one  degree  better  or  one  degree  worse. 

3.  In  any  invalid  mood  make  superior  form  three 
degrees  worse. 

Postulates:  Theorems: 

0  /  7)';  (7  /  a)';  (e  /  a)'.     The  other  (39)  invalid  moods. 

§11.  Having  now  completely  defined  the  relationships 
of  better  and  worse  by  deducing  all  the  prepositional  forms 
into  which  these  relationships  may  enter,  there  remains 
for  this  chapter  only  one  other  task,  which  is  to  deduce 
symbolically  the  moods  of  immediate  inference  and  the 
syllogism. 

§12.  From  the  postulates  and  principles,  which  are 
given  below,  all  the  moods,  valid  and  invalid,  of  immediate 
inference  may  be  deduced. 

Principles: 

i.     (y  /  x)  (x»  b  Z  xa  b)  Z  (ya  b  ^  ya  b). 
iv.  (xZ  y)  (yZ  z)Z  (xZ  z). 

Postulates  :  Theorems  : 

aab^  aba*,  jSabZ  /3ba;  €&b  Z.  eba.       The  other  valid  moods. 


A  PRIMER  OF  LOGIC  21 

Principles:* 

ii.     (x  /  y)  (xa  b  /  xa  b)  Z  (xa  b  Z  ya  b)' 
(x//y)  (xa  b  ^  xa  b)  Z  (xa  b  Z  ya  b)' 

iii.     (y  /  x)  (xa  b  /  xa  b)  Z  (xa  b  Z  ya  b)' 
(y//x)  (xa  b  /  xa  b)  Z  (xa  b  Z  ya  b)' 

v.     (xZ  y)    (xZ  z)'Z  (yZ  z)' 
(xZ  z)'  (yZ  z)  Z  (xZ  y)' 

Postulates:  Theorems: 

(dab^  cab)';  (eabZ  aab)';  (Tab^  7ba)'.          The  other  invalid 

moods. 

§13.  All  the  valid  and  invalid  moods  of  the  syllogism 
may  be  deduced  from  the  assumptions  which  follow. 
The  right  to  convert  simply  in  any  form  but  7  is  ex- 
pressed under  v  and  vi.  It  will  be  evident  that  some  of 
the  postulates  might  have  been  saved  at  the  expense  of 
introducing  new  principles,  and  conversely.  The  first 
two  principles  for  the  deduction  of  the  invalid  moods 
under  iv  are  theorems  from  the  ones  that  have  gone  be- 
fore under  i,  by  (xy  Z  z)  Z  (xz'  Z  y'). 

Principles: 

i.     (y    /  x)  (xba  zbc  /  xca)  Z  (yba  zbc  Z  yca) 
(y    /  x)  (zab  xcb  Z  xca)  Z  (zab  ycb  ^  yca) 

v.  (xy  Z  z)  (z  Z  w)  Z  (xy  Z  w) 
(xy  Z  z)  (w  Z  x)  Z  (wy  Z  z) 
(xy  Z  z)  (w  Z  y)  Z  (xw  Z  z) 

*These  principles,  except  the  second  under  iii  are  really 
special  cases  of  principles  i  and  iii  under  the  syllogism,  obtained 
from  the  latter  by  making  b  =  c,  the  primed  part  of  the  ante- 
cedent in  iii  becoming  unprimed  in  the  special  case. 


22  A  PRIMER  OF  LOGIC 


Postulates:     (aaa)z;     (777)  r» 

Theorems:    The  other  (26)  valid  moods. 

Principles: 

ii.  (y  /  x)  (z  I  y)  (xyZ  z)'  Z  (xz  Z  y)' 
(y  /  x)  (z//y)  (xyZz)'Z(xzZ  y)' 
(y//x)  (y  /  z)  (xyZz)'Z(xzZy)' 
(y///x)  (y  /  z)  (xyZ  z)'Z  (xzZ  y)' 
(y  /x)  (z  /  y)  (xyZz)'Z(zyZx)' 
(y  /  x)  (z//y)  (xyZz)'Z(zyZx)' 
(y//x)  (y  /  z)  (xyZz)/Z(zyZx)/ 
(y///x)  (y  /  z)  (xyZ  z)'Z  (zyZ  x)'. 

iii.  (x  /  y)  (xa,b  xb,c  Z  xca)'  Z  (xa,b  xb,c  Z  yea)' 
(x  //  y)  (xa,b  xbtC  Z  Xca)'  Z  (xa,b  xb,c  Z  yca)' 
(y  /  x)  (xa,b  xb,o  Z  xcay  Z  (xa,b  xb,c  Z  yca)' 

iv.  (x  /  z)  (yab  xcb  Z  Xca)'  Z  (yab  zcb  Z  zca)' 
(x  /  z)  (xba  ybc  Z  xca)'  Z  (zba  ybc  Z  zca)' 
(y  /  x)  (xab  xbc  Z  xca)'  Z  (yab  ybc  Z  yca)' 

vi.*  (xy  Z  z)'  (w  Z  z)  Z  (xy  Z  w)' 
(xy  Z  z)'  (x  Z  w)  Z  (wy  Z  z)' 
(xy  Z  z)'  (y  Z  w)  Z  (xw  Z  z)' 


(aetf)',    (a/37)'t    (ae7)'x 

(aa7)'x    (ajSe)',    (^a7)'x    (jSeyX,    (777)'2    (e^7)'x    (e7e)'3 

(aac)'z    (a77)'3    (jSjSe)',    (7a7)'2    (7c7)' 


The  other  (206)  invalid  moods. 


*Principles  v  and  vi  are  of  course  not  independent.  The 
first  under  v  is  a  variation  of  transitivity,  the  third  a  variation 
of  the  second  by  xyZ  yx.  Those  under  vi  follow  from  those 
under  v  by  (xy  Z  z)  Z  (xz'  Z  y')  .  Principles  v  under  immediate 
inference  follow  from  transitivity  by  the  same  principle. 


A  PRIMER  OF  LOGIC  23 

§14.  We  have  already  pointed  out,  (note  p.  2),  that 
the  product  7a,b  ea,b  does  not  vanish  in  general  if  we  allow 
the  possibility  of  the  limiting  values  0  and  1  for  the  terms. 
Under  these  conditions,  (7ee)2,4  and  (ej€)I>2  are  not  valid 
moods  of  the  syllogism,  for  they  become  7<>i  e0tl/.  0,  for 
a  =  c  =0  and  b  =  1.  A  logic,  which  recognizes  these  limiting 
values  of  the  terms,  will  have  to  postulate  (eye)'^  say, 
which  yields,  (eye)'2  and  (yee) '2,4. 

The  only  change,  which  we  should  then  have  to  make 
in  chapters  II  and  III,  would  be  to  replace  (777) '2  among 
the  postulates  by  (eye)*!,  from  which  (777) '2  follows  as  a 
theorem,  and  to  subtract  (eye)It2  and  (yee)2,4  from  the  list 
of  valid  moods. 

This  logic,  which  might  be  called  non-Aristotelian,  or 
semi-Aristotelian,  or  imaginary  logic,  is  more  general  than 
the  ordinary  or  classical  logic  and  includes  the  latter  as 
a  special  case,  becoming,  in  fact,  identical  with  it  when  the 
field  of  its  application  is  narrowed  so  as  to  exclude  ''nothing" 
and  "universe"  as  limiting  values  of  the  terms.  One 
principle,  which  is  true  in  the  special,  but  not  in  the  general 
case,  is: 

(y   /  x)  (xba  zcb  /  xca)  Z  (vba  zcb  Z  yca), 

and  this  principle  may  be  regarded  as  the  differentiating 
character  of  the  two  cases.  If  we  had  chosen  to  assume 
it,  instead  of  the  first  principle  under  i,  we  could  have  saved 
the  third  postulate,  but  the  second  principle  under  iv 
would  not  then  have  followed  as  a  theorem. 

The  definitions  of  chapter  I,  §3,  hold  for  both  cases; 
the  only  change  to  be  made  in  implications  V  in  order  to 
make  them  true  in  the  more  general  logic,  will  be  to  replace 
7aa  =0  by  7aa4:  0,  and  this  property  has  not  been  made  use 
of  in  defining  the  7-form, 


24  A  PRIMER  OF  LOGIC 

The  Aristotelian  forms,  A,  E,  I,  O,  (see  Appendix  I), 
will  yield  only  eight  valid  moods  of  the  syllogism,  under  the 
new  condition,  instead  of  the  twenty-four  valid  moods 
commonly  recognized.  They  satisfy  all  the  conditions  of 
maximum  simplicity  in  the  special  or  classical  instance — 
they  are  the  best  possible  forms  to  choose  for  the  con- 
struction of  an  Aristotelian  logic — but  they  fail  in  the  gen- 
eral instance,  for  they  then  lose  their  peculiar  advantage, 
that,  corresponding  to  any  member  of  the  set  there  should 
be  another  member  of  the  set  which  represents  its  contra- 
dictory. 


A  PRIMER  OF  LOGIC  25 


EXERCISES 

By  the  aid  of  the  principles, 

(xy  Z  z)  (z  Z  w)  Z  (xy  Z  w) 
(xy  Z  z)  (w  Z  x)  Z  (wy  Z  z) 
(xy  Z  z)  (w  Z  y)  Z  (xw  Z  z) 

we  are  enabled  to  convert  in  either  premise  or  the  conclusion. 
The  example  which  follows  will  illustrate  the  method. 

(7ba  <*cb  Z  Tea)    (o%Z   Otj\c)  Z   (7ba  «bc  ^  Tea) 


(1)  From  (eye)x  derive  (7€e)2,4. 

(2)  From  the  principles  and  the  postulates  in  the  text  de- 
duce the  remaining  valid  moods. 

(3)  From  the  postulates,  (aaa)lt  (777)1,  (OTY)I  and  the  prin- 
ciple (y/x)  (xba  zcb  Z  xca)  Z  (yba  zcb  ^  yca)  deduce  the  remaining 
valid  moods. 

If  we  identify  the  subject  and  predicate  of  the  conclusion 
in  the  mood,  (Pafi)3,  we  obtain  |3ba«baZ  0,  (chapter  I,  impli- 
cations v).  By  the  aid  of  (xyZ  0)  Z  (xZ  y')  it  follows  that 

«abZ  /3'ab- 

(4)  Deduce  as  many  as  possible  of  the  propositions  of  the 
form,  xabZ  y'ab,  (chapter  I,  p.  2)   by  identifying  subject  and 
predicate  in  the  conclusion  of  the  valid  moods  of  the  syllogism. 

The  principles, 

(xyZ  z)'(wZ  z)Z  (wyZ  w)' 
(xyZ  z)'(xZ  w)Z  (wyZ  z)' 
(xyZ  z)'(yZ  w)  Z  (xwZ  z)' 

enable  us  to  convert  in  either  premise  or  the  conclusion. 

(5)  From  (fi8/3)'x,  derive  the  invalidity  of  this  mood  in  the 
other  figures. 

(6)  From  (e/3e)'i  alone  and  principles  iii,  iv  and  vi  deduce 
seventy  other  invalid  moods. 


(7)  From  (ajfryX,  alone  and  principles  ii  deduce  nineteen 
other  invalid  moods, 


26  A  PRIMER  OF  LOGIC 

(8)  Deduce  the  invalid  moods  in  the  third  figure,  whose 
conclusion  is  in  the  7-form. 

(9)  Deduce  the  invalid  moods  in  the  fourth  figure,  whose 
major  premise  is  in  the  7-form. 

(10)  From  7771  alone,  deduce  forty-six  valid  implications 
of  the  form  La,b  Mb.c  £  N'ca,  —  L,  M  and  N  representing  only 
the  unprimed  letters. 

(11)  Assuming    eyelt2  and  yee2t4  to  be  invalid,  show  that 
777!  yields  only  thirteen  valid  implications  of  the  form  given 
in  the  last  exercise. 


(12)  Assuming  (/3ftJ')'i  (B^')\  (/S/Se7)'.  (770')'=  (rtfO'j  (777')'. 

(77*')'*  (erf%  («rO'i  ("0')'.  («O'. 

deduce  sixty-nine  other  non-implications  of  the  same  form. 

Any  non-implication  of  the  form,  Lb,a  Mc,b  ^  N'ca,  which 
contains  an  a-form  may  be  proven  invalid  by  identifying  terms 
in  the  a-form.  Thus  /3ba  Pbc  £  oi  c&  reduces  to  /3ba  /  /3'ba  for 
c  =  a;  7baacb^  7'ca  reduces  to  7baZ  7rba  for  c  =  b,  etc. 


(13)  Establish  the  invalidity  of  the  thirty-four  non-impli- 
cations of  the  form  Lb,a  Ma,b/  N'ca  not  accounted  for  in  the 
preceding  exercise. 

(14)  Show  that  there  are  thirty-six,   and  only  thirty-six, 
distinct  valid   implications   of   the   form   La>b  Mb,c  Nc,aZ  0,  — 
L,  M  and  N  representing  only  the  unprimed  letters,  a,  /3,  7,  e. 

(15)  Derive  indirectly  the  (777')  'i  of  exercise  12. 

A  certain  number  of  the  postulates  for  the  derivation  of  the 
invalid  moods  of  the  syllogism  (p.  21)  may  be  shown  indirectly 
to  be  invalid  by  reducing  them  to  invalid  moods  of  immediate 
inference.  Thus  (aa/3)i  reduces  to  a^Z.  c/ab  when  the  terms 
in  the  conclusion  are  identified,  and  (a&y)  i  reduces  to  /3ca^  Tea, 
when  we  identify  terms  in  the  major  premise  and  suppress  the 
part  a&&  (see  chap.  IV). 

(16)  Establish  the  invalidity  of 


(€#€)!          («  €7)  i 

(copy)!  (0:77)3  (707) 


A  PRIMER  OF  LOGIC  27 

Most  of  the  other  postulates  for  the  deduction  of  the  invalid 
moods  of  the  syllogism  may  be  reduced  by  the  method  of  the 
following  example: 

Suppose  that  (epy)i  is  valid.  Now  (e/3y')i  is  valid  by  a 
preceding  exercise. 

•  *  •  (  €ba  &b  Z  Tea)    (  €ba  0cb  Z  T'CB)  Z    (*ba  fob  ^   0) 

since  (xZ  y)  (xZ  y')  Z  (xZ  0). 


Consequently  €ba  0cb  ^  cca. 

If  now  we  postulate  (ej8e)'x,  it  follows  that 

(17)  Establish  the  invalidity  of 


(£77)3          (707)2          (777)4 

(7*7)2  (€77)3  (cea)! 

The  postulates  (Pey)'i  (ej8e)'x  and  (eye)'3  that  remain  (p.  21) 
may  be  reduced  by  the  following  method: 

(jSba  &b  /    e'ca)'   (|8ob  ^  Vcb)  ^  (^ba  7'cb  /    e'ca)' 

by  a  principle  under  vi  and  the  postulate  of  a  preceding  exercise 

GSba   7'cb/    OVtfba    tcaZTcb)' 

which  yields  (Pey)^  by  simple  conversion  in  the  major  premise. 
(18)  Establish  the  invalidity  of  (  c/3  e)z  and  (  ey  e)3. 

Any  non-implication  of  the  form  La,b  Mb,c  ^  Nca,  which 
contains  an  a-premise,  may  be  reduced  to  an  invalid  mood  of 
immediate  inference,  and  so  shown  to  be  invalid,  by  identifying 
terms  in  the  a-premise.  All  of  the  other  invalid  moods  may  be 
derived  from  the  postulates  of  exercise  (12),  the  forms  of  im- 
mediate implication  given  in  chapter  I,  the  principles  iv  and  vi 
of  chapter  III,  together  with  (xy  Z  z)  Z  (xz'Z  y')  and  (xy  Z  z)  Z 
(z'y^  A 

19.  From  the  postulates  of  exercise  (12)   deduce  all  the 
non-implications  of  the  form  La,b  Mb,0^  Nca,  without  making 
use  of  principles  ii  and  iii  of  this  chapter. 

20.  Show    that  there    exist  no  valid  implications  of  the 
form  L;a,b    Mb,c  Z  Nca  or  La,b  MVC  ^  Nca  and    consequently 
none  of  the  form  L;a,b  M'b.c  Z  NQa  or  L'a,b  M^.o  Z  N'ca. 


CHAPTER    IV 

§15.  The  sorites  is  a  form  of  implication  of  the 
general  type:* 

Xx(i|2)  x2(2,3)  x3(3f4)—  xn-x(n7T,  n)  Z  xn(n  i), 

in  which  the  number  of  terms  is  greater  than  three. 

Certain  valid  moods  of  the  sorites  can  be  constructed 
from  chains  of  valid  syllogisms.  Thus  the  chain  of  syl- 
logisms: 

a(i,2)  0(2,3)  Z  a(3i), 

a(3i)  0(3,4)  Z  a(4i), 

0(41)  0(4,5)  Z  a(si), 

will  yield  a  valid  sorites,  viz : 

0(1,2)  0(2,3)  0(3,4)  0(4,5)  Z  0(51),   for 

{ 0(1,2)  0(2,3)  Z  0(31)  }  Z  {  0(1,2)  0(2,3)  0(3,4)  Z  0(31)  0(3,4) } 

.'.     0(1,2)  0(2,3)  0(3,4)  Z  0(41),  by  the  second  syllogism  and 

the  principle  of  transitivity. 

{  0(1,2)  0(2,3)  0(3,4)  Z  0(41)  }  Z 

{  0(1,2)  0(2,3)  0(3,4)  0(4,5)  Z  0(41)  0(4,5)  } 

.  *.     0(1,2)  0(2,3)  0(3,4)  0(4,5)  Z  o(si),  as  before. 

Consequently  in  general,  if 

XI(l,2)  Xa(2.3)  Z  X3(3l) 
X3(3l)  X4(3,4>  Z  Xs(«) 
Xs(4l)  X6(4,5)  Z  X7(si) 


X2n-s(n-i  i)  X2n.4(n-i,  n)  Z  X2n-3(ni) 


*In  this  chapter  it  will  be  more  convenient  to  employ  the 
notation  x(ab)  for  xab  or  x(i,2)  for  xlf2.  The  comma  between 
the  terms  means  that  the  term  order  is  not  settled. 


A  PRIMER  OF  LOGIC  29 

be  a  chain  of  valid  syllogisms,  then 

xz(i,2)  x2(2|3)  x4(3,4)—  x2n_4(iiT7,  n)  Z  x2n.3(ni) 

is  a  valid  mood  of  the  sorites.  It  remains  to  be  proven 
that  the  only  valid  moods  that  exist  can  be  constructed 
from  chains  of  valid  syllogisms.  The  proof  depends  on 
the  following  principles. 

Principle  i. — A  valid  mood  of  the  sorites,  which  has 
one  premise  of  the  same  form  as  the  conclusion,  will  re- 
main valid,  when  as  many  of  the  other  premises  as  we 
desire  are  put  in  the  a-form. 

Principle  ii. — A  valid  mood  of  the  sorites  will  remain 
valid,  when  as  many  terms  have  been  identified  as  we  desire. 

Principle  iii. — An  a-premise,  whose  subject  and  predi- 
cate are  identical,  may  be  suppressed  as  a  unit  multiplier. 

Principle  iv. — A  valid  mood  of  the  sorites,  none  of 
whose  premises  has  the  same  form  as  the  conclusion,  will 
remain  valid,  when  as  many  premises  as  we  desire  are  put 
in  the  a-form. 

Theorem  i. — There  exists  no  valid  mood  of  the  sorites, 
in  which  none  of  the  premises  has  the  same  form  as  the 
conclusion. 

For  (principle  iv)  put  all  the  premises  after  the  first 
in  the  a-form.  Then  by  identifying  terms  (principle  ii) 
the  mood  of  the  sorites  can  be  reduced  (principle  iii)  to  an 
invalid  syllogism  of  the  form: 

Xx(l,2)   a(2,3)  Z   Xn(3l). 

Conclusion  in  the  a-form. 

At  least  one  of  the  premises  is  in  the  a-form  (theorem  i). 
If  one  of  the  remaining  premises,  xr(s  -  i,  s),  be  not  in  the 
a-form,  put  each  one  of  the  other  premises  in  the  a-form, 


30  A  PRIMER  OF  LOGIC 

if  all  but  xr  be  not  already  in  that  form  (principle  i). 
Then  by  identifying  terms  (principle  ii)  the  mood  of  the 
sorites  will  reduce  (principle  iii)  to  an  invalid  syllogism 
of  the  form: 


xr(s  -  i,  s  )  a(s,  s  + 1)  Z  a(s  + 1  s  -  i), 
or  a(s  -  2,  s  -  i)  xr(s  -  i,  s)  Z  a(s  s  -  2). 

Consequently  all  the  premises  are  in  the  a-form  if  the 
mood  of  the  sorites  is  valid  and  the  sorites  is  of  the  general 
type: 

0(1,2)  01(2,3) —  a(n-  i,  n)  Z  a(ni), 

which  can  be  constructed  from  the  chain  of   valid  syllo- 
gisms : 

a(2,i)  a(3,2)  Z  a(3i), 

a(3i)  01(4,3)  Z  a(4i), 

a(4i)  01(5,4)  Z  a(si), 


a(n  - 1  i)  a(n,  n  -  i)  Z  a(n  i). 

Conclusion  in  the  /3-form. 

At  least  one  premise,  xt,  is  in  the  /3-form  (theorem  i), 
and  all  the  other  premises  are  in  the  a-form.  For  suppose 
one  of  the  other  premises  xr  (s  - 1,  s)  were  not  in  the  a-form. 
Put  all  the  premises  (principle  i)  except  xt  and  xr  in  the 
a-form.  Then  by  identifying  terms  (principle  ii)  the  mood 
of  the  sorites  will  be  reducible  to  an  invalid  syllogism 
(principle  iii)  of  the  form: 


0(s-  i,  3-2)  xr(s,  s-  i)  Z  |8(s  s-  2), 
or  xr(s,  sT~7)  j8(s,  s  +  i)  Z  P(s~+i    s-  i). 


Consequently  the  sorites  must  be  of  the  form: 
0(2,3)  —  a(s,  s  -  i)  /3(s  +  i»   s)  a(s  +  i,   5+2)  —  a(n-  i,    n) 


A  PRIMER  OF  LOGIC  31 

Z  |8(n  i),  which  can  be  constructed  from  the  chain  of 
syllogisms : 

0(1,2)  0(2,3)  Z  a(ai) 
a(ai)  0(3,4)  Z  0(41) 


a(s  -  i  i)  a(s  -  i,  s)  Z  a(s  i  ) 
a(si)  j8(s, 


|8(sTi  i)  o(sTii  s+2  )  Z  0(s+2  i) 


0(n-i  i)  a(n-i,  n)  Z  #(n  i). 

Conclusion  in  the  y-form. 

At  least  one  of  the  premises  is  in  the  7- form  (theorem  i). 
Each  7-form  in  the  antecedent  must  present  its  terms  in 
the  order  (s  s  - 1).  For  suppose  that  y(s  - 1  s)  should 
appear  as  one  of  the  premises.  Put  each  one  of  the 
remaining  premises  in  the  a-form  (principle  i).  Then  by 
identifying  terms  (principle  ii)  the  sorites  will  reduce  to 
an  invalid  syllogism  (principle  iii)  of  the  form: 


T(S  -  i  s)  a(s,  s  +1)  Z  y(s  +1  s  - 


or         a(s  -  i,  s  -  2)  T(S  - 1  s)  Z  y(s  s  -  2). 

Pursuing  the  same  reasoning  as  before  it  can  be  shown 
that  no  |8-  or  e-  premises  can  occur.  One  form  of  this 
sorites  may  consequently  be  7(21)—  7(11  n- 1)  Z  7(111),  which 
can,  in  fact,  be  constructed  from  the  chain  of  valid  syl- 
logisms: 

7(21)  7(32)  Z  7(31) 

7(31)  7(43)  Z  7(41) 

7(n~M~  0  7(n  iiTi)  Z  7(11  i). 

All  the  other  forms  of  valid  sorites  with  a  7-conclusion 
are  obtained  from  the  above  type  by  transforming  one  or 


32  A  PRIMER  OF  LOGIC 

more  of  the  premises  into  the  a-form  in  every  possible 
way  under  the  restrictions  of  theorem  i.  Each  one  of 
these  types  can  be  built  up  from  a  chain  of  valid  syllo- 
gisms each  member  of  which  has  one  of  the  forms : 

(77T)i,  (a77)i,2,  or  (7017)  i, 3- 

Conclusion  in  the   e-form. 

At  least  one  of  the  premises  is  in  the  e-form  (theorem  i), 
and  there  is  not  more  than  one  e-premise.  For,  if  there 
are  two  or  more  e-premises,  put  all  the  premises  but  two 
of  the  e-premises  in  the  a-form  (principle  i).  Then  by 
identifying  terms  (principle  ii)  we  will  come  upon  an  invalid 
syllogism  (principle  iii)  of  the  form: 

e(s  -  i,  s)    e(s,  s+i)Z   e(s  +  i    s  -  i). 

There  can  be  present  no  /3-premise.  For  suppose 
xr  (s,  s  -  l)  to  be  a  /3-premise.  Put  all  the  premises  except 
xr  and  the  e-premise  in  the  a-form  (principle  i).  By  identi- 
fying terms  (principle  ii)  we  will  come  upon  an  invalid 
syllogism  (principle  iii)  of  the  form: 


/3(s,  s-  i)  e(s,  s+i)  Ze(s  +  i  s-  i), 
or          e(s  -  i,  s  -  2)  /3(s,  s  -  i)  /  e  (s  s  -  2). 

Any  7-premise  coming  after  the  e-premise  must  pre- 
sent its  terms  in  the  order  (s  s  -  i).  For  suppose  7(5,  s  -  i) 
coming  after  the  e-premise  to  present  the  term  order 
(s-  i  s).  Put  all  the  premises  except  y(s  -  i  s)  and  the 
e-premise  in  the  a-form  (principle  i).  Then  by  identifying 
terms  (principle  ii)  we  will  come  upon  an  invalid  syllogism 
(principle  iii)  of  the  form: 


e(s  -1,5-2)  7(s  -  i  s)  Z   e(s  s  -  2). 

Any  7-premise   coming   before   the    e-premise   must 
present  its  terms  in  the  order  (s  - 1  s).  For  suppose  7(3,  s  -  i) 


A  PRIMER  OF  LOGIC  33 

coming  before  the  e-premise  to  present  the  term  order 
(s  s  -  i).  Put  all  the  premises  except  7(5  s  -  i)  and  the 
c-premise  in  the  a-form  (principle  i).  Then  by  identifying 
terms  (principle  ii)  we  will  come  upon  an  invalid  syllogism 
(principle  iii)  of  the  form  : 

T(S  s-  i)  c(s,  s  +  i)  Z   e( 


One  form  of  this  sorites  may  be,  consequently, 


7(12)  7(23)— 7(3  -  2  «T~i)  e(s,  s  -  i)  7(s+i  s)— 7(n  n~^7)  Z 
e(n  i),  which  can,  in  fact,  be  constructed  from  the  chain 
of  valid  syllogisms: 

7(12)  7(23)  Z  7(13) 
7(13)  7(34)  Z  7(14) 


7(l  S  -  2)  7(s  -  2    S  -  l)  Z   7(1  S  -  l) 

7(1  s  -  i)  e(s,  s-  i)  Z   c(s  i) 
c(s  i)  7(s+7  s)  Z   e(s~+T  0 

e(n-  i  i)  7(n  n  -  1)  Z   e(n  i) 

All  the  other  forms  of  valid  sorites  with  an  c-conclusion 
are  obtained  from  the  above  type  by  replacing  one  or  more 
of  the  7-forms  by  a-forms  in  every  possible  way.  Each 
new  type  can  be  constructed  from  a  chain  of  valid  syllo- 
gisms, each  member  of  which  has  one  of  the  forms: 


There  exist,  consequently,  no  valid  moods  of  the  sorites 
which  can  not  be  constructed  from  chains  of  valid  syllo- 


gisms.* 


*If  (7€c)2,4  and  (eye)If2  are  to  be  regarded  as  invalid  moods, 
(see  the  concluding  remarks  of  chapter  III),  then  it  can  be  shown 
at  once  that  no  7-premise  can  occur  when  the  conclusion  is  in 
the  e-form.  The  general  form  of  such  a  sorites  will  be, 


34  A  PRIMER  OF  LOGIC 


EXERCISES 

(1)  Construct  a  valid  sorites  from  the  chain  of  valid  syllo- 
gisms : 

a2i732^  73i» 

73i<*43^   74i, 

74x  ?54  ^  7si- 

(2)  By  the  aid  of  the  principles  of  chapter  IV,  reduce  the 
valid  sorites,  a21  y32  a43  y54  Z  7SI,  successively  to  each  one  of  the 
three  valid  syllogisms  of  example  1. 

(3)  Prove  the  invalidity  of  the  sorites, 

7zi  73*  734  754  ^  7Si- 

(4)  From  what  chain  of  valid  syllogisms  can  the  sorites, 
«i,2723  €3,47s4«6f5^  «6i  be  constructed? 

(5)  If  (€7e)i.2  and  (7€e)2,4  be  regarded  as  invalid  moods  of 
the  syllogism,  (see  the  concluding  remarks  of  chap.  Ill),  prove 
the  invalidity  of  the  sorites, 

7i2  7a3"7s-a  s-i    €  s,  s-i  7s+i  s~7n  n-i  Z    €nr 


a(i,2)a(2t3) — a(s  -  i.s)e  (s,  s  +  i)a(s+i,  8+2) — a(n-  i,  n)/  e(ni) 
which  can  be  built  up  from  the  chain  of  syllogisms, 

a(i,2)a(2,3)Z  a(si) 

a(3i)a(s,4)Z  a(4i) 


a(s  -  i  i)a(s  -  i,  s)Z  a(si) 
a(si)€(s,  sTi)  Z  e(s+i  i) 
c(s  +  i  i)  a  (s-H,  s+2)Z  e(s+2 


c(n  -  i  i  )a  (n  -  1 1  n)  Z  c  (n  i) 


APPENDIX  I 

On  the  Simplification  of  Categorical  Expression  and  the 
Reduction  of  the  Syllogistic  Figures 

If  a  and  b  represent  classes,  there  are  four  ways  in 
which  they  may  be  related  categorically,  the  one  standing 
for  subject,  the  other  for  predicate.  These  four  forms  of 
relationship  are  always  represented  by  the  letters,  A, 
E,  I,  O,  i.  e. 

Aab=all  a  is  b, 
Eab=no  a  is  b, 
Iab  =  some  a  is  b, 
Oab=not  all  a  is  b. 

Historical  efforts  have,  been  made  to  reduce  the  num- 
ber of  these  relationships.  If  symbols  be  invented  to 
denote  some  a  (a)  and  not-a  (ax),  the  last  three  may  be 
represented  by  means  of  the  first,  for: 

Eab=Aabi>   Iab=Aab>  Oab=Aabi- 

But  an  essential  difference  is  here  left  undistinguished 
and  the  number  of  necessary  forms  will  not  have  been 
reduced  by  this  device.  If  a  new  symbol  be  employed  for 
all  a  (a)  and  another  for  the  copula  is  (  Z  ),  we  shall  have: 

Aab=aZ  b, 
Eab=aZ  bi, 
lab  =  aZb, 
Oab=aZ  bx. 

The  four  separate  categorical  forms  have,  accordingly, 
been  gotten  rid  of  at  the  cost  of  introducing  four  new  unde- 
fined symbols,  so  that  no  economy  of  our  indefinables  has 
been  effected. 


36  A  PRIMER  OF  LOGIC 

It  is  to  be  observed  that  the  word  some,  which  is  im- 
plicit or  explicit  in  the  meaning  of  part  of  each  proposition, 
means  some  at  least,  possibly  all.  Another  set  of  proposi- 
tions, in  which  some  is  to  mean  some  at  least,  not  all,  may 
be  used  to  replace  the  traditional  ones.  These  other  forms 
are: 

aab  =  all  a  is  all  b, 

/3ab=some  a  is  some  b, 

Tab  =  all  a  is  some  b, 

€ab=no  a  is  b. 

In  addition  we  shall  have  to  employ 
the  hypothetical  form, 

xab  /  y&b  =xab  implies  yab, 
{  xab  ^  yab}r  =  xab  does  not  imply  yab, 

the  conjunctive  form, 

Xab  '  yab  =  xab  and  yab, 

the  disjunctive  form, 


Xab 

Each  member  of  the  set,  A,  E,  I,  O,  may  be  expressed 
in  the  members  of  the  set,  a,  /3,  7,  c,  and  conversely,  so 
that  the  two  are,  in  fact,  logically  equivalent,  although 
each  one  has  certain  advantages  peculiar  to  itself. 

The  members  of  the  second  set  have  this  property, 
that,  if  one  is  true,  then  all  the  others  are  false.*  We 
assume,  accordingly,  the 

Postulates:       a&b  L  /3'ab       0ab  ^  T'ab      Tab  /  T'ba 
dab  £  T'ab       jSab  £    c'ab 
«ab  ^    c'ab       Tab  ^     c'ab 

*Provided  we  exclude  the  limiting  values  0  and  1  for  a  and  b. 
The  ordinary  definitions  of  these  limits  allow  70i  and  c0i  be  true 
together. 


A  PRIMER  OF  LOGIC  37 

from  which  follow,  by  the  principle  of  the  denial  of  the 
consequent,  the 

Theorems:         eab  /  a'ab         Tab  ^  o/ab 

Cab  /    0'ab  Tab  /    0'ab 

Cab  /   T'ab  /3ab  ^    a'ab 

Consequently* 

aab'/3ab=0  /?ab'Tab=0  Tab*Tba=0 

ttab    *7ab=0  0ab   *    Cab=0 

aab  *  cab=0         Tab  '  €ab=0  I 

From  the  Definitions:** 

Aab=aab  +  Tab 

Eab  =  Cab 

lab    =  ttab  +  )3ab  +  Tab  +  Tba  II 

Oab  =  €ab  +  /3ab  +  Tba 

we  obtain  immediately*** 

dab  =  Aab   "  Aba 
/3ab  =  lab   *  Oab    '  Oba 
Tab  =Aab   "  Oba 
Cab  =  Eab 

It  is  an  advantage  of  the  forms  of  the  original  set, 
an  advantage  which  the  set,  a,  /3,  T>  c,  does  not  possess, 
that  the  contradictory  of  any  letter  is  represented  by  a 
single  other  letter  of  the  set.  Suppose  that  we  were  to 

*a  /3  =  0  reads:    a(is  true)  and  fi(is  true )  is  impossible. 

**A,  E,  I,  O  are  simply  the  sums  given  in  equations  II. 
That  they  are  the  traditional  Aristotelian  forms,  is  only  an 
accident  of  the  reader's  application.  Hence  equations  II  are 
definitions  and  not  postulates. 

***Multiplying  out  the  sums  in  II  as  if  they  were  ordinary 
polynomials,  applying  the  results  of  I,  and  assuming  that  a,  J3 
and  c  are  simply  convertible. 


38 


A  PRIMER  OF  LOGIC 


combine  this  advantage  with  that  of  simple  convertibility 
in  a  new  set  of  forms. 

To  do  this  it  would  seem  to  be  enough  to  subtract 
from  the  meaning  of  Aab  the  part  7&b,  (equations  II),  and 
to  add  this  part  to  the  meaning  of  Oab.*  Our  new  set  of 
forms  becomes: 


HI 


=  Cab 
liab    =Ctab  + 


+  Tab  +  7ba 
+  Tab  +  7ba 

the  analogues  of  the  old  letters  being  represented  by  the 
corresponding  Greek  vowels. 

From  equations  I  and  III,  and  remembering  that  the 
sum  of  dab,  /3ab,  7ab>  cab,  Tba  makes  up  the  propositional 
"universe,"  the  results  of  the  following  table,  yielding  all 
the  moods  of  immediate  inference,  will  easily  be  seen  to 
hold. 


True 

Implies  the 
truth  of 
only 

Implies  the 
falsity  of 
only 

False 

Implies  the 
truth  of 
only 

Implies  the 
falsity  of 
only 

a 

a,  i 

e,  0 

a 

0 

a 

e 

e,  0 

a,  i 

€ 

L 

e 

i 

i 

€ 

L 

€,  0 

a,  t 

o 

o 

a 

0 

a,  i 

€,   0 

An  induction  of  these  results  shows  that  a  =  o't 
o  =  af,  e  =  if,  i=  e',  and  that,  consequently,  contradictory 

*Here  would  seem  to  be  another  instance  of  the  manner  in 
which  the  language  of  symbols  may  free  a  science  from  the 
accidents  imposed  upon  its  development  by  the  language  of 
speech.  The  last  two  members  of  the  new  set  have  apparently 
no  simple  verbal  expression. 


A  PRIMER  OF  LOGIC 


a, 


pairs  are    a,    o  and   c,  t.     Likewise   contraries  are 
subcontraries  are  i,  0  ;  subalterns  are  a,  t  and  €,  o. 

If  we  define  an  affirmative  form  as  one  that  becomes 
unity  when  subject  and  predicate  have  been  identified 
and  a  negative  form  as  one  that  becomes  unity  when  sub- 
ject and  predicate  have  been  made  contradictory,  then  it 
is  a  result  of  the  following 

Postulates:*       aaa    is  a  true  proposition, 


Theorems: 


18', 


»> 


>» 


»        M  » 

»        M  >» 

>»        M  » 


Qt  » 

P  aa 


>»  » 


/  »       >»  »  M 

T  aa 

and  equations  III,  that  a  and  t  are  affirmative  and  that  e 
and  o  are  negative  forms. 

If  a  distributed  term  be  one  modified  by  the  quanti- 
tative adjective  all,  it  will  be  seen  that  a  and  c  distribute 
both  subject  and  predicate,  while  i  and  o  distribute  neither. 
These  results  are  summarized  in  the  table  below,  the 
distributed  terms  being  underlined. 


Affirmative 

Negative 

,f 

Universal 

dab 

€ab 

Particular 

tab 

0ab 

*x  is  true  is  to  be  represented  by  x=l,  x  is  false  by  x  =  0. 
(See  Boole,  Investigation  of  the  Laws  of  Thought,  ch.  XI,  p. 
169).  The  theorems  follow  by  equations  I,  and  equations  III 
become  as  a  result  of  them,  aaa=l,  caa  =  0,  iaa  =  l,  0aa  =  0, 
«aa  =  0,  6aa  =  l,  iaa  =  0, 0aa=L  Employing  the  usual  notation, 
a^not-a. 


40  A  PRIMER  OF  LOGIC 

The  traditional  rules  for  detecting  the  invalid  moods 
of  the  old  syllogism,  constructed  from  the  set  A,  E,  J,  O, 
hold  for  the  new  syllogism,  built  up  out  of  the  forms, 
a,  e,  i,  o.  These  rules  are: 

1.  Two  negative  premises  do  not  imply  a  conclusion. 

EX.     €bn   €cb    Z    Cca» 

2.  Two  affirmative  premises  do  not  imply  a  nega- 
tive conclusion.    Ex.  aba  acb  Z  eca. 

3.  An  affirmative  and  a  negative  premise  do  not 
imply  an  affirmative  conclusion.    Ex.  aba  «cb  ^  ac&. 

4.  Two  premises,  in  neither  of    which  the  middle 
term   is  distributed,   do  not  imply  a  conclusion. 

EX.    tba  tcb  ^    tea. 

5.  Two  premises,  in  which  a  given  term  occurs  un- 
distributed, do  not  imply  a  conclusion,  in  which  that  same 
term  is  distributed.    Ex.  aba  tcb  Z  aca. 

The  valid  moods  which  remain,  and  which  of  course 
are  valid  in  all  four  figures,  since  each  one  of  the  forms 
is  simply  convertible,  are  twelve  in  number,  viz: 

aaa          aat          aee          aeo 

au  aoo  eae  eao 

tat  OQ.O          eio          teo 

It  has  been  previously  observed,  (note  p.  2),  that 
equations  I  hold  generally  only  when  the  limits  0  and  1 
are  excluded  as  possible  values  of  a  and  b.  If  these  pos- 
sibilities he  included,  we  shall  have  to  assume: 

{Tab  Z  e'ab  }'  and  .*.  {  eab  Z  7'ab  }',  since  TOI  eOI^  0. 
Equations  I  then  become: 

Ctab   '  0ab=0  |8ab   '7ab=0 

0  j8ab    '    Cab=0  IV 


A  PRIMER  OF  LOGIC  41 

Under  these  conditions,  the  fact  which  the  old  logic 
always  took  for  granted,  that  Eab  is  the  contradictory  of 
Iab,  and  Aab  the  contradictory  of  Oab,  no  longer  holds 
true.  For,  while  the  sum  of  each  of  these  two  pairs  of 
forms  is  the  prepositional  universe,  their  product  is  not 
the  prepositional  null,  (equations  II,  IV).  In  order  that 
Eab  *  lab  and  Aab  *  Oab  shall  vanish  for  all  values  of  the 
terms,  it  will  be  necessary  to  exclude  Tab  eab  from  the 
product.  A  new  set  of  forms,  in  which  part  of  the  meaning 
of  tab  is  subtracted  from  iab  and  added  to  eab,  will  satisfy 
this  requirement.  Let  this  new  set  be: 


C2ab  =  €ab  +  Tab  +  Tba 


02ab  =  €ab  +  Pab  +  Tab  +  Tba 

If  a,  e,  i,  o,  be  replaced  by  a2,  c2,  i2,  o2,  respectively 
in  the  table,  (p.  38),  all  the  results  of  such  a  new  tabulation 
will  be  seen  to  hold,  (equations  IV,  V).  The  same  defini- 
tions as  given  before  will  make  a2  and  i2  affirmative,  e2  and 
o2  negative  forms,  (equations  V,  and  the  postulates  and 
theorems,  p.  39),  but  since  e2  distributes  neither  subject 
nor  predicate,  e2  i2  o2  and  i2  e2  o2  will  not  be  found  among 
the  valid  moods  of  the  syllogism,  (see  p.  40).  The  same 
rules  (p.  40)  for  the  detection  of  the  invalid  moods  will 
hold  for  the  new  syllogism,  but  rule  1  is  now  redundant, 
being  a  corollary  of  rule  4. 

It  might  perhaps  appear  that  our  original  symmetry, 
(that  of  equations  I),  which  was  interrupted  by  the  neces- 
sity of  allowing  TOI  to  stand  as  a  true  proposition,  could 
be  saved  by  assuming  that  the  null  class  exhausts  no  part 
of  the  universe,  i.  e.  all  of  nothing  is  some  of  everything, 
might  be  regarded  as  a  false  proposition,  Now  A0»  =  l, 


42  A  PRIMER  OF  LOGIC 

or  cioa-f  7oa  =  l,  is  Schroder's  definition  of  the  null  class, 
and  Aoa  will  be  a  true  proposition  for  all  values  of  a  if 
7oi  be  true,  whereas,  7oi=0  involves  A0i=0.  These  con- 
sequences lead  us  to  the  alternatives  of  either  giving  up 
our  symmetry,  (in  equations  I),  or  else  of  regarding  the 
null  class  as  not  essential  to  our  algebra. 

It  is  finally  to  be  noted — what  was  obvious  in  the 
beginning — that,  while  the  members  of  the  set,  a,  e,  t,  0, 
can  be  expressed  in  the  members  of  the  set,  a,  0,  7,  e, 
the  latter  can  not  be  expressed  in  the  former.  Conse- 
quently, an  essential  difference  has  been  lost,  and  the 
existence  of  a  completed  logic  of  the  new  forms  would  not 
put  aside  the  necessity  of  working  out  the  logic  of  the  old. 

The  attempts  of  the  logician  to  discover  a  set  of 
categorical  forms,  which  establish  a  complete  symmetry 
among  the  moods  and  figures  of  the  syllogism,  are  as  old 
as  the  science  itself.  The  end  would  be  attained  if  a  new 
set  could  be  selected  so  as  to  satisfy  the  following  con- 
ditions: 

1.  Each  form  of  the  set  must  be  simply  convertible. 

2.  Corresponding  to  any  member  of  the  set,  there 
must  occur  another  which  represents  its  contradictory. 

3.  The  new  set  must  yield  at  least  one  valid  mood 
of  the  syllogism. 

4.  Each  member  of  the  new  set  must  be  represent- 
able  in  the  members  of  a  set  already  proved  necessary 
and  sufficient  to  express  all  differences,  (the  set  A,  E,  I, 
O,  say),  and  conversely. 

If  Xab  be  any  categorical  form,  the  simplest  functions 
of  x,  which  are  themselves  categorical  and  which  are  in 
general  simply  convertible,  are  xab  *  xba  and  xab  +  xba.  It 
will  be  enough,  therefore,  in  order  to  satisfy  condition  1, 


A  PRIMER  OF  LOGIC  43 

to  assume  as  a  new  set  of  forms  either  such  a  sum  or 
such  a  product  of  each  one  of  the  old  forms  (A,  E,  I,  O, 
say). 

The  equation,  {xab  *  xba  }'  =x'ab  +  x'ba,  suggests  at 
once  what  our  manner  of  satisfying  condition  2  must  be, 
for  since  the  product,  xab  '  Xba  is  the  contradictory  of  the 
sum  x'ab  +  x'ba  and  x  the  contradictory  of  x',  if  xab  +  xba 
[respectively  xab  '  xba]  be  chosen  as  one  of  our  new  forms, 
x'ab  '  x'ba  [respectively  x'ab  +  x'ba]  must  be  chosen  as  one 
of  the  others. 

Remembering  that 


lab    =Iab     *  Iba    =Iab     +  Iba, 

and  that    Eab  =  rab  and  Iab  =E'ab, 

it  follows  that  our  choice  of  a  new  set  of  forms  is  limited 
to  the  following  two: 

AI=Aab    'Aba,  Ex=Eab,  (1) 

Oba,  Ix    =  Iab, 


A2=Aab  +  Aba,  E2=Eab,  (2) 

O2=Oab   'Oba,  I2    =    lab- 

It  will  be  found,  however,  that  the  set  (2)  yields  no 
valid  moods  of  the  syllogism.  Consequently,  applying 
condition  3,  our  choice  is  seen  to  be  unambiguously  re- 
stricted to  set  (1),  which  yields  twelve  moods,  valid  each 
one  in  each  of  the  four  figures.  These  will  be  found  to 
be,  (dropping  the  subscripts): 

A  A  A,  A  A  I,  AEE,  A  E  O, 
All,  AGO,  EAE,  EAO, 
I  A  I,  OAO,  EIO,  IEO. 

It  will  be  impossible,  however,  to  satisfy  condition  4, 
since  every  expression  involving  the  new  forms  will  be 


44  A  PRIMER  OF  LOGIC 

simply  convertible.  Consequently,  an  essential  difference 
has  been  left  undistinguished,  and  it  will  not  be  possible 
to  substitute  the  new  forms  for  the  old.  The  new  forms 
are,  in  fact,  identical  with  ar,  ex,  tx,  olt  considered  above. 
From  this  latter  discussion  and  from  the  discussion 
that  has  gone  before,  we  conclude,  that,  if  it  be  necessary 
to  retain  in  our  set  of  forms  at  least  one  that  is  not  simply 
convertible,  it  will  be  impossible  to  satisfy  the  condition 
2  above,  unless  the  null-and  one-class  be  excluded  or  de- 
nned in  some  way  other  than  ordinary. 


APPENDIX  II 

Historical  Note  on  De  Morgan's  New  Prepositional  Forms 

In  introducing  the  notion  of  contradictory  terms  into 
logic  De  Morgan  discovered  two  new  prepositional  forms, 
which  cannot  be  directly  expressed  by  means  of  the  A,  E, 
I,  O  relations  of  traditional  logic.*  Suppose  that  we  de- 
note these  two  forms  by  U  and  V,  i.  e. 

U  (ab)  =A11  not  a  is  b, 

V  (ab)  =Some  not  a  is  not  b, 

A  (ab)  =A11  a  is  b, 

E  (ab)  =No  a  is  b, 

I    (ab)  =Some  a  is  b, 

O  (ab)  =Some  a  is  not  b. 

U  and  V  are  simply  convertible,  for  (if  a  =  not-a) 
U(ab)  =A  (ab)  =A  (ba)  =U  (ba),  (converting  in  A  by  the 
principle  of  contraposition),  and 

V  (ab)  =  I  (ab)  =  I  (ba)  =  V  (ba),  (converting  simply  in  I). 

V  distributes  both  subject  and  predicate  while  U 
distributes  neither,  for  V(ab)  =O(ab)  =O(ba)  (converting 
in  O  by  contraposition)  and,  since  O  distributes  its 
predicate,  both  a  and  b  are  distributed  terms;  similarly 
U(ab)  =A(ab)  =A(ba)  (converting  in  A  by  contraposition) 
and,  since  A  does  not  distribute  its  predicate,  a  and 
b  are  undistributed  terms. 

If  an  affirmative  form  be  defined  as  one  that  becomes 
the  subcontrary  of  itself  when  the  subject  and  predicate 

*Formal  Logic,  p.  61. 


46 


A  PRIMER  OF  LOGIC 


have  been  identified,  and  a  negative  form  as  one  that 
becomes  the  contrary  of  itself  under  the  same  conditions, 
it  will  be  seen  that  A,  I  and  V  are  affirmative,  that  E,  O 
and  U  are  negative  forms. 

These  results  are  summarized  in  the  following  table, 
the  distributed  terms  being  underlined. 


Affirmative 

Negative 

Universal 

A(ab) 

E(ab) 

Particular 

I(ab) 

0(ab) 

Indefinite 

V(ab) 

U(ab) 

Below  are  tabulated  all  the  forms  of  immediate  im- 
plication which  hold  among  the  six  propositions  A,  E,  I, 
O,  U,  V.  The  (144)  implications  and  non-implications 
necessary  to  establish  unambiguously  the  results  of  the 
table  can  be  derived  from  a  certain  number  of  postulates 
and  the  commonly  assumed  principles  of  traditional  logic. 


True 

Implies 
falsity  of 

Implies 
truth  of 

False 

Implies 
falsity  of 

Implies 
truth  of 

A 

E,  0,  U. 

A,  I,  V. 

A 

A. 

O. 

E 

A,  I. 

E,  O. 

E 

E. 

I. 

I 

E. 

I. 

I 

A,  I. 

E,  0. 

O 

A. 

O. 

0 

E,  0,  U. 

A,  I,  V. 

U 

A,V. 

0,  U. 

U 

U. 

V. 

V 

U. 

V. 

V 

A,V. 

O,  U. 

An  induction  of  these  results  will  show  that: 
Contradictory  pairs  are:    A,  O;  E,  I;  U,  V; 
Contrary  pairs  are:    A,  E;  A,  U; 


A  PRIMER  o'LtGJe'  •' V:     f       ;  ,  ,      47 

Subcontrary  pairs  are:     I,  O;  O,  V; 

Subalternate  pairs  are:    A,  I;  A,  V;  I,  V;  I,  U;  E,  O; 

E,  U;  E,  V;  U,  O. 

In  order  to  show  how  the  new  forms  fit  the  ancient 
scheme  and  as  an  illustration  of  method  let  us  solve  the 
array  of  the  syllogism.  We  first  observe  (see  table)  that 
A  weakens  ambiguously  to  I  or  V;  that  O  strengthens  am- 
biguously to  E  or  U.* 

Rules: 

1.  In  any  valid  mood  interchange  either  premise 
and  the  conclusion  and  replace  each  by  its  contradictory. 

2.  In  any  valid  mood  strengthen  a  premise  or  weaken 
a  conclusion. 

3.  In  any  valid  mood  convert  simply  in  any  form 
but  A  or  O. 

Postulates: 

A  A  A  (1st  figure)  is  a  valid  mood. 

AUU  (  "  "    )  "  " 

EAE  (  "  »    )  »  »»      » 

UEA  (  "  »    )  >'  »      » 

From  these  rules  and  postulates  will  follow  sixteen 
valid  moods  in  the  1st  figure,  twenty  in  each  one  of  the 
2nd  and  3rd  figures,  and  twenty-one  in  the  4th  figure. 

In  order  to  deduce  the  invalid  moods  let  us  assume 
the 

Rules: 

1.  In  any  invalid  mood  interchange  either  premise 
and  the  conclusion  and  replace  each  by  its  contradictory. 

*If  x  implies  y  but  y  does  not  imply  x,  then  x  is  said  to  be  a 
strengthened  form  of  y,  and  y  is  said  to  be  a  weakened  form  of  x. 


48  -  'A'PRfMER  OF  LOGIC 

2.  In    any    invalid    mood    weaken    a    premise   or 
strengthen  a  conclusion. 

3.  In  any  invalid  mood  convert  simply  in  any  form 
but  A  or  O ;  and 

Postulates: 

A  A  A  (4th  figure)   is  an  invalid  mood. 

AAO  (  "  "  )  "  " 

AAV  (3rd  "  )  "  " 

AAI  (2nd  "  )  "  " 

AAO  (1st  "  )  "  " 

EEI ( "  "  ) "  " 

EEO  (  "  "  )  "  " 

EU A  (  "  "  )  "  " 

EUO  (  "  "  )  "  " 

UUO  (  "  "  )  "  " 

U  U  V  (  "  "  )  "  "    "    M 

From  these  postulates  and  rules  follow  the  remaining 
(772)  invalid  moods. 

It  will  be  seen  at  once  that  the  rules  of  the  old  logic 
for  the  immediate  detection  of  the  invalid  moods  of  the 
syllogism  no  longer  hold.  To  give  only  one  illustration: 
A  term  may  appear  distributed  in  the  conclusion  of  a  valid 
mood  and  be  undistributed  in  the  premise. 


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